Strong coincidence and Overlap coincidence

TL;DR

This paper establishes a connection between strong coincidences and overlap coincidences in Pisot substitution tilings, providing a converse to previous work and clarifying their relationship under certain conditions.

Contribution

It proves the equivalence of strong and overlap coincidences for certain Pisot substitution tilings, extending the understanding of their relationship.

Findings

Strong coincidences are equivalent to overlap coincidences under specified conditions.

The result applies to degree ≥ 2 Pisot substitutions with topological constraints.

Provides a converse to Akiyama-Lee's work on coincidences.

Abstract

We show that strong coincidences of a certain many choices of control points are equivalent to overlap coincidence for the suspension tiling of Pisot substitution. The result is valid for degree $\ge 2$ as well, under certain topological conditions. This result gives a converse of the paper by Akiyama-Lee and elucidates the tight relationship between two coincidences.